Finite Group Databasebeta

← #46825664

Subgroups — #46825664

Order 34432.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575499C211yesmaximal, characteristic, complemented
2#31771413C218yesmaximal, complemented
3#31771408C218yesmaximal, complemented
4#31771403C216yesmaximal, complemented
5C2×#22575478C216yesmaximal, complemented
6#31771372C211yesmaximal, characteristic, complemented
7#585×C269C211yesmaximal, characteristic, complemented
8#22575499—24no—
9#22575499C2218yescomplemented
10C2×#16864403—21no—
11C2×#16864402C2213yescomplemented
12C22×D1076C2213yescomplemented
13C2×D4×D269—23no—
14#22575484C22112yescomplemented
15#22575483C2214yescomplemented
16#22575479C22112yescomplemented
17#22575478C22112yescomplemented
18#22575478C22124yescomplemented
19#22575473C22124yescomplemented
20C2×#16864389C22112yescomplemented
21C2×#16864387C22112yescomplemented
22#22575460C2218yescomplemented
23C2×#16864386C2216yescomplemented
242−1+4×C269—21no—
252−1+4×C269C2214yescomplemented
26H(5,𝔽2)×C269—23no—
27H(5,𝔽2)×C269C2214yescomplemented
28C2×#41×C269—24no—
29C2×#41×C269C2211yescharacteristic
30C2×#41×C269C2216yescomplemented
31C23×D269—23no—
32#16864403—28no—
33#16864402—26no—
34#16864402C23112yes—
35C2×Dic538C2311yescharacteristic
36C2×D1076—23no—
37C2×D1076—26no—
38C2×D1076C23112yes—
39C2×D1076C2313yes—
40C2×C22⋊D269—23no—
41D4×D269—224no—
42C2×C4×D269—23no—
43#16864392C2318yescomplemented
44#16864390C2318yescomplemented
45#16864389C23124yescomplemented
46#16864387C23124yescomplemented
47#16864386C2314yes—
48#16864386C23112yescomplemented
49C2×#14524130C23112yescomplemented
50#41×C269—24no—
51#41×C269—212no—
52#41×C269—26no—
53#41×C269—26no—
54#41×C269D414yes—
55#41×C269C2314yes—
56#41×C269C2314yes—
57#41×C269C23112yescomplemented
58C2×Q8×C269—24no—
59C2×Q8×C269D411yescharacteristic
60C2×Q8×C269C2316yescomplemented
61C2×D4×C269—26no—
62C2×D4×C269—212no—
63C2×D4×C269D413yes—
64C2×D4×C269C2316yescomplemented
65C22×C4×C269—26no—
66C22×C4×C269C2313yes—
67C22×D269—23no—
68C22×D269—212no—
69C22×D269—86no—
70Dic538—21no—
71Dic538C2412yes—
72D1076—212no—
73D1076—23no—
74D1076C2416yes—
75C22⋊D269—212no—
76C2×Dic269—21no—
77C4×D269—212no—
78#14524130C2418yes—
79C23×C269—26no—
80C23×C269C2×D413yes—
81Q8×C269—26no—
82Q8×C269C2×D414yes—
83Q8×C269C2414yes—
84D4×C269—24no—
85D4×C269—212no—
86D4×C269—26no—
87D4×C269C2×D4112yes—
88D4×C269C2414yes—
89C2×C4×C269—26no—
90C2×C4×C269—24no—
91C2×C4×C269—212no—
92C2×C4×C269C2×D416yes—
93C2×C4×C269C2×D413yes—
94C2×C4×C269C2411yescharacteristic
95C2×C4×C269C2416yes—
96C22×C269—24no—
97C22×C269—43no—
98C22×C269—86no—
99C22×C269C22×D416yes—
100C22×C269C22×D411yescharacteristic
101C2×D269—26no—
102C2×D269—46no—
103C2×D269—812no—
104C4×C269—24no—
105C4×C269C22×D411yescharacteristic
106C4×C269C22×D416yes—
107C4×C269C2511yescharacteristic
108Dic269—22no—
109C2×C269—21no—
110C2×C269—26no—
111C2×C269—44no—
112C2×C269C23×D411yescharacteristic
113D269—46no—
114C269#358511yescharacteristic, complemented, Sylow
115#3585—2691nomaximal, Sylow
116#585D26911yescharacteristic, complemented
117C2×H(5,𝔽2)—2691no—
118#577—2698no—
119#576—2698no—
120#575—2696no—
121C2×AGL(1,ℤ/8)—2696no—
122#568—2691no—
1232−1+4—21no—
1242−1+4C2×D26914yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—2698no—
127H(5,𝔽2)—5384no—
128H(5,𝔽2)C2×D26914yescomplemented
129C2×#41—24no—
130C2×#41—2693no—
131C2×#41—5381no—
132C2×#41C2×D26911yescharacteristic
133C2×#41C2×D26916yescomplemented
134C22×D4—2693no—
135C22×D4—5383no—
136#137—2694no—
137#137—26912no—
138AGL(1,ℤ/8)—26912no—
139AGL(1,ℤ/8)—26924no—
140AGL(1,ℤ/8)—26912no—
141#135—26924no—
142C2×SD16—26912no—
143C2×D8—26912no—
144#131—2698no—
145C2×M16—2696no—
146C24—5383no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—26912no—
152#41—5388no—
153#41—5386no—
154#41C22⋊D26914yes—
155#41C22×D26914yes—
156#41C22×D26914yes—
157#41C22×D269112yescomplemented
158C2×Q8—24no—
159C2×Q8—2691no—
160C2×Q8C22⋊D26911yescharacteristic
161C2×Q8C22×D26916yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—26912no—
165C2×D4—2693no—
166C2×D4—5383no—
167C2×D4—53824no—
168C2×D4—5383no—
169C2×D4—5386no—
170C2×D4C22⋊D26913yes—
171C2×D4C22×D26916yescomplemented
172C22×C4—26no—
173C22×C4—5383no—
174C22×C4C22×D26913yes—
175Q16—2698no—
176SD16—26924no—
177SD16—2698no—
178D8—26924no—
179M16—2694no—
180M16—26912no—
181C2×C8—26912no—
182C23—26no—
183C23—5383no—
184C23—53812no—
185C23—21526no—
186C23C2×C22⋊D26913yes—
187Q8—26no—
188Q8—2692no—
189Q8—5381no—
190Q8C2×C22⋊D26914yes—
191Q8C23×D26914yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—2696no—
196D4—53812no—
197D4—5383no—
198D4—53812no—
199D4C2×C22⋊D269112yes—
200D4C23×D26914yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—5381no—
205C2×C4—53812no—
206C2×C4C2×C22⋊D26916yes—
207C2×C4C2×C22⋊D26913yes—
208C2×C4C23×D26916yes—
209C2×C4C23×D26911yescharacteristic
210C8—2698no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—5386no—
215C22—10766no—
216C22—215212no—
217C22C22×C22⋊D26916yes—
218C22C22×C22⋊D26911yescharacteristic
219C4—24no—
220C4—5382no—
221C4C22×C22⋊D26911yescharacteristic
222C4C22×C22⋊D26916yes—
223C4C24×D26911yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—10766no—
228C2C23×C22⋊D26911yescharacteristic
subgroupG/H
centerC2C23×C22⋊D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
Fitting#585×C269C2
socleC2×C269C23×D4
hypercenter#585D269
hypocenterC269#3585
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. #31771413C2semidirect
3. #31771408C2semidirect
4. #31771403C2semidirect
5. C2×#22575478C2semidirect
6. #31771372C2semidirect
7. #585×C269C2semidirect
9. #22575499C22semidirect
11. C2×#16864402C22semidirect
12. C22×D1076C22semidirect
14. #22575484C22semidirect
15. #22575483C22semidirect
16. #22575479C22semidirect
17. #22575478C22semidirect
18. #22575478C22semidirect
19. #22575473C22semidirect
20. C2×#16864389C22semidirect
21. C2×#16864387C22semidirect
22. #22575460C22semidirect
23. C2×#16864386C22semidirect
25. 2−1+4×C269C22semidirect
27. H(5,𝔽2)×C269C22semidirect
30. C2×#41×C269C22semidirect
43. #16864392C23semidirect
44. #16864390C23semidirect
45. #16864389C23semidirect
46. #16864387C23semidirect
48. #16864386C23semidirect
49. C2×#14524130C23semidirect
57. #41×C269C23semidirect
60. C2×Q8×C269C23semidirect
64. C2×D4×C269C23semidirect
114. C269#3585semidirect
116. #585D269semidirect
124. 2−1+4C2×D269semidirect
128. H(5,𝔽2)C2×D269semidirect
133. C2×#41C2×D269semidirect
157. #41C22×D269semidirect
161. C2×Q8C22×D269semidirect
171. C2×D4C22×D269semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225754992
#317714132
#317714082
#317714032
C2×#225754782
#317713722
#585×C2692
#3585269